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Theorem dfrnf 4880
Description: Definition of range, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 14-Aug-1995.) (Revised by Mario Carneiro, 15-Oct-2016.)
Hypotheses
Ref Expression
dfrnf.1 𝑥𝐴
dfrnf.2 𝑦𝐴
Assertion
Ref Expression
dfrnf ran 𝐴 = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦}
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝐴(𝑥,𝑦)

Proof of Theorem dfrnf
Dummy variables 𝑤 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfrn2 4827 . 2 ran 𝐴 = {𝑤 ∣ ∃𝑣 𝑣𝐴𝑤}
2 nfcv 2329 . . . . 5 𝑥𝑣
3 dfrnf.1 . . . . 5 𝑥𝐴
4 nfcv 2329 . . . . 5 𝑥𝑤
52, 3, 4nfbr 4061 . . . 4 𝑥 𝑣𝐴𝑤
6 nfv 1538 . . . 4 𝑣 𝑥𝐴𝑤
7 breq1 4018 . . . 4 (𝑣 = 𝑥 → (𝑣𝐴𝑤𝑥𝐴𝑤))
85, 6, 7cbvex 1766 . . 3 (∃𝑣 𝑣𝐴𝑤 ↔ ∃𝑥 𝑥𝐴𝑤)
98abbii 2303 . 2 {𝑤 ∣ ∃𝑣 𝑣𝐴𝑤} = {𝑤 ∣ ∃𝑥 𝑥𝐴𝑤}
10 nfcv 2329 . . . . 5 𝑦𝑥
11 dfrnf.2 . . . . 5 𝑦𝐴
12 nfcv 2329 . . . . 5 𝑦𝑤
1310, 11, 12nfbr 4061 . . . 4 𝑦 𝑥𝐴𝑤
1413nfex 1647 . . 3 𝑦𝑥 𝑥𝐴𝑤
15 nfv 1538 . . 3 𝑤𝑥 𝑥𝐴𝑦
16 breq2 4019 . . . 4 (𝑤 = 𝑦 → (𝑥𝐴𝑤𝑥𝐴𝑦))
1716exbidv 1835 . . 3 (𝑤 = 𝑦 → (∃𝑥 𝑥𝐴𝑤 ↔ ∃𝑥 𝑥𝐴𝑦))
1814, 15, 17cbvab 2311 . 2 {𝑤 ∣ ∃𝑥 𝑥𝐴𝑤} = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦}
191, 9, 183eqtri 2212 1 ran 𝐴 = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦}
Colors of variables: wff set class
Syntax hints:   = wceq 1363  wex 1502  {cab 2173  wnfc 2316   class class class wbr 4015  ran crn 4639
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-v 2751  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-br 4016  df-opab 4077  df-cnv 4646  df-dm 4648  df-rn 4649
This theorem is referenced by:  rnopab  4886
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